The Archimedean Property

Real Analysis · Axiom Academy

LESSON The Archimedean Property No real number is infinitely large or infinitely small — take enough steps, and the natural numbers leave any bound behind. Pick any two positive reals: a step size x and a target y . The Archimedean property promises a natural number n that pushes the multiple nx past y — even when x is minute and y is enormous. In the figure below, x = 0.7 and the target is y = 2.5 . Laying down copies of x , the total clears y for the first time at n = 4 , since 4(0.7) = 2.8 > 2.5 . The property wears two familiar disguises. Reach toward with the whole numbers, or reach toward 0 with their reciprocals — both are just nx > y with the right choice of x and y . For every there is an with n > r . (Take x = 1 , y = r : then .) For every there is an with . (Take , y = 1 : then , so .) For r = 4.2 , the first natural past it is n = 5 . For , the sequence first drops below at n = 5 , since is not yet less than but . 3. Why It's True — from Completeness The property is not an extra assumption; it follows from the completeness axiom : every nonempty set of reals that is bounded above has a least upper bound (a supremum). We argue by contradiction. You've seen the Archimedean property stated, worn as its two everyday faces, and proved from completeness — the property that makes limits, density, and approximation possible in . Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Real Analysis course.